in Space


The Euler Number


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Topology is often described as 'rubber sheet' geometry. This is because topology is about the general overall properties of shapes that remain unchanged as the shape is stretched and distorted. The shape can be stretched in any way imaginable, but as long as it is not torn, then it is regarded as topologically the same object. To a topologist a sphere is exactly equivalent to an ellipsoid, as is a cube, dodecahedron or other convex polyhedron. (Convex polyhedra are polyhedra whose faces do not penetrate the interior of the polyhedron.)

Metrical properties, such as the distance between two points on a surface, clearly do not interest topologists, as these properties can change dramatically when the surface is stretched. What topologists want to find are mathematical properties that do not change when the shape is stretched.

The most famous of these topological invariants, as they are called, was discovered by the great 18th Century Swiss mathematician Leonhard Euler (pronounced Oiler). He found a number, now known as the Euler number, that remains the same for all convex polyhedra. This number is simply F - E + V, where F is the number of face, E is the number of edges and V is the number of vertices of the polyhedron.

For instance, a cube has 6 faces, 12 edges and 8 vertices, so its Euler number is 2. Likewise, a dodecahedron has 12 faces, 30 edges and 20 vertices, so its Euler number is also 2. In fact, all convex polyhedra have an Euler number of 2.

No amount of stretching a sphere will transform its shape into a tyre or torus. The torus has a hole through it, topologists refer to the number of holes in a surface as the genus of the surface. So a torus has genus one, whereas a sphere has genus zero. If we examine a polyhedral representation of a torus, as shown in the animation above; this polyhedron has 16 faces, 32 edges and 16 vertices, so its Euler number, which is 16 - 32 + 16, equals zero. The Euler number is different to the Euler number of a convex polyhedron.

If we take two copies of the torus-shaped polyhedron, remove an outer face from each polyhedron and join them together along their open edges we can produce a polyhedron with genus two. We can now work out the Euler number of the new polyhedron.

The Euler number of the two toroidal polyhedra was twice zero, which is zero. But we discarded two faces, so this gives us an Euler number of minus two. (We also merged four edges and four vertices when we merged the two tori together, but this cancels out because both E and V are decreased by four.) The Euler number of a genus two surface is therefore -2. In general increasing the genus by one decreases the Euler number by two.

Text and animations by Nick Mee from the CD-ROM 'Life, the Universe and Mathematics' published by Virtual Image.